Can you run your own instance of archive.ph? Obviously not their software (they don't release it), but is there a version you can run yourself?
The important parts of what archive.ph provides are:
1. Headless chromium that works around 99.9% of the "headless fingerprinter" techniques. Yes, they do actually spin up a full browser engine; that's what sets them apart.
2. Decent curated collection of paywall-bypass tactics.
#2 is definitely available from other sources. So really what we're looking for is a one-click "headless chromium that deals with 99.9% of the fingerprint detectors".
Web archival is a very, very complex and deep topic. If you're interested in it, the archiveteam's wiki is a good resource. Start here, I think:
https://wiki.archiveteam.org/index.php/Introduction
I'm not terribly motivated right now to dive into the community and ask for help, so I won't seriously complain, but it's at least not trivial to use.
https://github.com/webrecorder/archiveweb.page
I also convert web snippets I find useful to the markdown format and store it in my Joplin notebook, so that way it lives on even if the website is gone.